Sincere question: could it be that the underlying structure of the universe is really simple but since we have no idea what it is we have to use exotic mathematics for it?
In fact that’s pretty much the story of the development of physics. It turns out Newtonian mechanics is emergent from Relativity. Maxwells equations are emergent from quantum mechanics. The behaviour of bosons is emergent from the behaviour of quarks.
As I understand it there are theoretical reasons to suspect that quarks do not have any decomposition though. We’ll see.
However relativity is far more complex than Newtonian physics.
Which is more complex depends on what you mean. There are fewer laws in the possibility space of generally-relativistic physics than Newtonian physics. So, which metric is more important? Less pleasant calculations or a larger search space?
Your comment suggests you weigh the calculational simplicity more heavily, but most physicists would come down on the other side of the issue.
What's your reason for saying this? I'm more on the mathematical side of things so I don't know that many physicists. But it's been my understanding that equations that are extremely difficult to solve, impossible to solve except numerically, or involve infinities that can't be explained, are a major pain point for physicists. I mean, that's the entire premise of this article.
- I'm a professional physicist and most people I know professionally think this way. People like symmetry, it helps clarify things, simplify things, provides powerful principles. If the cost is practical difficulties, well, that's just the cost of doing business; the physical understanding offered by simpler rules is beneficial. I know at least 2 people mentioned in the article would agree with that.
- I have no conceptual problem saying that the only solutions are numerical in nature if the principles are clear. Nobody promised physics should be easy. In fact some of the people mentioned in the article also have shown how their formal understanding might unlock better numerical methods!
- Some infinities are worse than others, and a modern effective field theory perspective makes me not worry about most examples.* With a Wilsonian understanding, renormalization is perfectly simple to understand. For theories which have perturbative UV fixed points you can formulate a lattice discretization which flows to that fixed point and you never encounter any infinity along the way.
Theories without a perturbative UV fixed point, well, that's where the trouble lies. Either there is no UV fixed point, in which case the theory is not valid for all energy scales and the troubling divergences point to an energy scale beyond which your theory is invalid. Or there IS a UV fixed point but it can only be found nonperturbatively.
Handed a QFT with no perturbative UV fixed point, how should you decide?
Well, one step back: should you, as a physicist, care?
For instance, why should we worry whether QED as a standalone theory is UV-complete? We know that in the real world electrodynamics mixes with the weak force at high energy. So whether QED as a standalone theory is UV-complete is a question that I'm not worried about. It is an interesting mathematical question, and that can only be answered with new techniques, such as resurgence. For standalone QED it's a question of pure mathematics, as far as anyone can tell. That's what these tools are good for, at the moment.
HOWEVER. I do admire the program of trying to show that more quantum field theories even exist mathematically, beyond the handful that we already know (which tend to have exotic properties). That seems important to me. But if it's false that's ALSO extremely interesting, it suggests that there are additional principles that we ought to understand.
* except for gravity, where the divergences are SO bad that even the EFT approach has problems.
I think you mean
> The behaviour of _hadrons_ is emergent from the behaviour of quarks.
--
Anyway, back to your main idea, the problem is that most of the times the new underlaying theory is even worse than the original one
> Newtonian mechanics is emergent from Relativity
For Newtonian Mechanics you need only basic calculus. For General Relativity you need curvature, tensors, two definitions of derivatives and other nasty stuff. [I never saw the details, but it's in my todo list.]
> Maxwells equations are emergent from quantum mechanics
Well, electromagnetism is just the local gauge invariance of the U(1) group. The idea is very simple but each word in that sentence needs like one semester to be decoded. [I saw the calculations a long time ago. I don't remember the details, but I remember the general idea. I wrote a comment with a oversimplified version https://news.ycombinator.com/item?id=8189346 )
> there are theoretical reasons to suspect that quarks do not have any decomposition though.
My favorite reason is that for a classic small ball, the ratio of the magnetic moment to the inertia moment is 1 (once you fix some nasty details about units), but for an elementary quantum particle it is 2. For composite particles, there is no theoretical value. https://en.wikipedia.org/wiki/G-factor_(physics)
The experimental value for protons is 5.5 and for Neutrons is 3.8, that is not surprising because we are sure they are composed particles.
For Electrons and Muons it's slightly more than 2, but we understand that difference quite well (but not perfectly, and that is related to the main point of the article here).
I don't think it has been measured directly for Quarks, but my guess is that it's used in some parts of the calculation of some Feynman diagrams, and if it were very different from 2 someone would have noticed.
>most of the times the new underlaying theory is even worse than the original one
Oh quite. Macroscopic emergent behaviour is often apparently simpler than the underlying causes and is in some way a generalisation of them.
The emergence of complex traits from simple rules still requires exotic mathematics to describe macroscopically.
Until you hit all the quantum weirdness and then it’s all wave functions and probabilities. That maybe comes out of something simple as well.
"Going down" simply means identifying laws that are more universal in that they can underly models of different systems, ideally "any known system". Quantum weirdness isn't significantly harder mathematically than what came before (we don't have an objective measure of how "hard" some piece of math is), it's just harder to relate to everyday experience. It's similar to how we got used to "masses attract each other" or "things just keep moving in a straight line", which seemed ridiculous to most of Newton's contemporaries.
We absolutely have a way to measure how hard a piece of math is: computational complexity. And quantum mecanichs is more computationally complex than newtonian mechanics (while general relativity is significantly harder still than both of them).
Is there a formal version of this claim somewhere? (Beyond theorems about quantum computers)
So, we can represent a physical theory as some algorithm A(init_condition, time, precision) = final_condition. Let's call the Newtonian model N and the quantum mechanical model QM.
My claim (which I think is completely uncontroversial) is that for the same init_condition, time, precision, you will need more computational steps to arrive at the same final_condition using QM than using N. This is the very definition of computational complexity.
If you want instead to compute the asymptotic complexity, we can take either time, precision, or some combination of the two, to be the parameter in respect to witch we compute complexity as a function.
Overall, this fact is quite uncontroversial since it is widely assumed that a classical computer requires an exponential amount of time to simulate a quantum computer running the same algorithm, for some algorithms such as Shor's. This is of course not yet proven, but it is widely considered very likely to be true. In fact, if it turns out to be false, it is quite likely that it will also turn out quantum mechanics is in fact reducible to classical mechanics (since the difference in computational speed is due to the special features of quantum mechanics as compared to classical, particularly the complex probability values that it arrives at).
To talk about "number of computational steps" (which I have never seen referred to as "definition of computational complexity") you need to specify a computational model. The number and the comparison between two programs will depend upon it and you can get whatever result you want by choice of the model.
The conversion of theory -> model -> program is also not at all straightforward. The complexity strongly depends on the conversion. If you specify "best possible conversion" for model to program (which is probably the easiest part of the huge amount of things you need to define), you cannot prove anything about it anymore.
The complexity difference between asymptotic classical and quantum calculations is usually determined by its scaling in the "size of state space". Why do you focus on time and precision? I don't know for sure but it seems reasonable to me that the complexity with respect to those two might actually be the same for common system models. Regardless, you have to choose some parameters and you provided no arguments that this choice is in any way objective.
> In fact, if it turns out to be false, it is quite likely that it will also turn out quantum mechanics is in fact reducible to classical mechanics
Strongly disagree for any reasonable notion of "reducible". Quantum theory is different from classical in many qualitative and experimentally accessible ways (e.g. Bell inequalities). Computational complexity isn't everything.
To end, I want to further contest your defining complexity of a theory according to "length of computation to get model prediction". An interesting alternative would be "length of theory specification under optimal compression " (i.e., "Kolmogorov complexity of a theory"). This notion obviously also isn't rigorously defined yet but I think it's about as hard to definine as your notion. Neither of them seems obviously the "right one" to me in this context.
The original Maxwell theory of electromagnetism is about 10 rather involved equations. Maxwell-Heaviside form is 4 simpler equations. A formulation using differential 3-forms is 2 simple equations. A formulation using geometric algebra / Clifford algebra is one utterly simple equation.
This way, the nabla symbol is very helpful in turning a group of (usually) 3 related PDEs into one pretty understandable one. Same for vector and matrix forms of common 2D and 3D transforms. Once you understand how these symbols work (they have a very regular structure), you can think at a bit higher level, and juggle with transforms that won't fit in your head in the scalar form, requiring toilsome operations on paper (or equivalent).
Say, general relativity is hard as it is; without various notational tricks which abstract away some complexity, it would likely be completely unwieldy.
A change of notation is like a refactoring in programming: a good set of powerful and well-defined functions makes the code to solve a problem significantly easier to produce and to understand, and harder to make an error.
Every system of equations can be written as "A = 0" for a suitable definition of A. That doesn't make it simple.
First is the idea of simple. If something has a few very well defined rules that are understood in isolation, but whose emergent behavior is beyond our ability to define, is it simple? Conway's Game of Life is somewhat the default example. 2 very simple rules (or perhaps more, depending upon specifically how you count them), but it gives rise to a Turing complete system. Math itself is another example, as mathematicians seek to find simple rules from which math arises, yet even for the subsets of math that are limited to such rules, is it really fair to call it simple?
The second idea is that of an underlying structure. Does the universe have an underlying structure, and even if it does, does that exist in side of some more foreign concept? What happens before the big bang? Why did the big bang happen when it did? Are there other universes, both from the many worlds interpretation of quantum mechanics, and universes that entirely separate from our own. These seem questions that feel almost entirely in the realm of science fiction, not physics, but there are plenty of theoretical physicists who dive into this field even though it currently doesn't produce testable hypothesis and is thus outside the scope of proper science.
Well, I hope so. If reality was inconsistent and things happened arbitrarily with no rhyme or reason I think we’d be in big trouble.
I’m not totally unsympathetic to the view that maths is fundamental though. It’s an interesting way to think about it.
But from what I’ve read, the deeper you go, the more it’s difficult to find something other than mathematics. When you ask what is a particle you find out it’s probably an excitation of a quantum field. So what is a quantum field if not a mathematical structure? Is there a physical reality to wave function collapse?
Maybe it doesn’t matter. The shut up and calculate crowd doesn’t seem to care.
It is an interesting speculation, but it’s also possible it’s just confusing the map for the terrain.
It also seems like a comfortable answer to the question about about what's happening every time I cause wave function collapse and split the universe. I'm just creating a new mathematical structure.
Aren't you describing a classical field?
As I understand it, a quantum field is essentially varying probabilities (the wavefunction).
> they produce consequences in the world
They don't produce consequences in the world, they are the world. You and I, we're excitations of a quantum field. We're of the wave function. The quantum field encodes all the information that is us.
Nobody calls the Standard Model a law, for example. The modern view is that the Standard Model is a low-energy effective field theory.
But, whatever supplants the SM, we still expect the principle of translation invariance to hold.
Until, that is, we have evidence for a paradigm shift. If we discover physics that really can't be described, for example, by dynamics happening in a geometric space, then we'll have to give up that principle. Strongly-coupled stringy dynamics seems to have non-geometric phases, for example.
So our statement of laws is more a description of the current best paradigm (say, the operating system), rather than our best model (the program).
A system can be complete, meaning all true statements can be built based on axioms, but it cannot be sound leading to contradictions.
Alternatively a system may be sound, but not all true statements could be derived.
And finally there are statements impossible to prove because the proof is undecidable.
It’s an interesting question whether the physical world is perfectly or merely highly consistent. If it’s made of mathematics, it may be that it cannot be perfectly consistent. So if we ever find that it is perfectly consistent, that might be evidence that it isn’t made of mathematics.
Most likely we’ll never be able to tell, but we’ll see. Or at least maybe our descendants will.
What does that even mean? The physical world is not a formal system in any obvious sense.
Nor does "consistent" apply to mathematics, by the way, only to formal systems used/studied by mathematics. You cannot mathematically prove or disprove that mathematics is consistent or define precisely what that would mean because mathematics itself isn't rigorously defined. If you define it as "whatever mathematicians are doing", I guess it's in some sense inconsistent, since mathematicians often disagree.
On the other hand, if our universe is one of many arbitrary possible universes, I’d say it’s not a mathematical structure (although it could still be the only universe that exists, hypothetically).
Apart from that, I hope upon some reflection you realize that your sentence is an illustration of my point.
Of course, you don't help anyone make progress in physics just by suggesting they should "use some philosophy for a change".
You aren't really pointing it out, you're just claiming that without any evidence at all.
You should be able to back up your claim by showing some example (not fully worked out, just some idea) of a philosophy-inspired physical theory that is simpler than current theories but more or less as precise in its measurable predictions of what will happen next in sine physical system.